Shell corrections

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

There is, however, an exception in an infinite homogeneous electron gas, electrons are delocalized. Neglecting their orbital motion does not contradict quantum mechanics, and lacking localization is unproblematic when only cross 

Unlike for an atom, a shell correction expansion up to high orders is possible for an electron bound in a harmonic-oscillator potential [16]. However, this system is characterized by only one parameter and, hence, does not readily allow to separate kinetic from other contributions. 

Conceptionally the situation is much clearer in classical theory because the cross-over toward negative stopping numbers of the Bohr logarithm L ohi — In(Cmu /ZiUof) can easily be avoided [11]. Orbital motion can be incorporated into the initial conditions [17], although the actual evaluation in Ref. [17] was carried through only to the leading term in u .  

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Bethe s formula requires that the velocity of the incident particle be much larger than that of the atomic electrons, a condition not easily fulfilled by the K-electrons except in the lightest elements. The required correction, called the shell correction, is denoted by subtracting a quantity C from the stopping number. In the penetration of high-Z material, even L-shell correction may be required. In that case, C denotes the sum total of all shell corrections. The subject of shell correction has been extensively treated by several authors, and various graphs and formulas are available for its evaluation (see, e.g., Bethe andAshkin, 1953). 

Let us now invert the logic and define the Casimir energy as the energy resulting from the geometry-dependent part of the density of states (d.o.s.) - a concept that is closely related to the shell correction energy in nuclear physics ... 

A fairly general way to evaluate shell corrections is based on kinetic theory [1]. Here it is assumed that shell corrections account for orbital motion and nothing else. The theoretical basis is a relation between the stopping number Lq for a target at rest and the stopping number L for a moving target [1],... 

Fig. 2. Stopping cross section for hydrogen in argon. Calculated from binary theory with and without shell correction. Experimental data from numerous laboratories compiled in Ref. [6].

Figure 2 shows the stopping force of argon gas on protons. Experimental data are compared to two predictions of binary stopping theory to be discussed below, excluding and including shell correction, respectively. The difference is seen to be substantial. 

In the case of atomic stopping, the orbital implementation of the KT brought as a natural consequence the ability to calculate shell correction terms as [27,32]... 

Ions in crystals I expt. II Inner shell correction r. m. s. from outer Xm Median from X-ray ( )... 

From Table 3 the outermost electron-cloud radii of Cl and K+ from crystal data should be in the ratio 23.0/13.7 = 1.30. KCl is chosen for standard substance to minimise the two opposing sources of error uncertainty in the standard value for Li+ and uncertainty in the inner-shell correction. In KCl the structural nearest-neighbour distance is 3.147 A, hence Wasastjerna s Criterion so applied leads to the structural radii... 

A Consistent Calculation of Atomic Energy Shell Corrections Strutinsky s Method in the Hartree-Fock-Roothaan Scheme... 

One can rewrite the first-order shell-correction term given by Eqn (7b) by using the self-consistent energiese, and the above decomposition ... 

Equations (7) can be viewed as a formal Taylor-series expansion, around the averaged part of the one-particle density matrix, of the HF energy functional E[p] [16, 18], this defining a shell-correction series . In Eqn (13) the first-order term of this expansion is expressed in terms of the single-particle energies e,. 

With the determination of Yo the above-described averaging procedure can be performed, yielding the averaged value E p and the shell corrections S,E p and SjEjjp for the energy, as well as the averaged and fluctuating parts p and 5p of the one-particle density matrix. 

This process is repeated until self-consistency is reached, the final values p and n, being noted p and n, Using these values the first-order shell correction to the energy can be written ... 

In Section 3 we have formulated Strutinsky s shell-correction method in the framework of the analytic HFR scheme, for single open-shell atoms and molecules in their ground state. The consideration of two or many open-shell systems could be performed following the same pattern. Both the averaged part of the energy, E,jp, and its first-order shell-correction part, 8,E pr, have been derived in analytic form, and the self-consistent process for determining them has been described. 

We have performed calculations aimed at testing the applicability of Strutinsky s shell-correction method to atoms and also at investigating the main features of the irregular part 5,Ej of Ehfr as a function of the atomic number Z. For this... 

Equation (3) incorporates relativistic effects, effects of target density, and corrections to account for binding of inner-shell electrons, as well as the mean excitation energy C/Z is determined from the shell corrections, S/2 is the density correction, Ifj accounts for the maximum energy that can be transferred in a single collision with a free electron, m/M is the ratio of the electron mass to the projectile mass, and mc is the electron rest energy. If the value in the bracket in Eq. (4) is set to unity, the maximum energy transfer for protons... 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

To get more accurate stopping powers, two corrections are added to the basic stopping power formula. One is shell correction and the other is density elfect correction. Including these elfects, the formula will be written as follows [8,9] ... 

STRUTINSKY S SHELL-CORRECTION METHOD IN THE EXTENDED KOHN-SHAM SCHEME APPLICATION TO THE IONIZATION POTENTIAL, ELECTRON AFFINITY, ELECTRONEGATIVITY AND CHEMICAL HARDNESS OF ATOMS... 

Abstract. Calculations of the first-order shell corrections of the ionization potential, 6il, electron affinity, 5 A, electronegativity, ix, and chemical hardness. Sir] are performed for elements from B to Ca, using the previously described Strutinsky averaging procedure in the frame of the extended Kohn-Sham scheme. A good agreement with the experimental results is obtained, and the discrepancies appearing are discussed in terms of the approximations made. 

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